If we are to understand the nature of heavenly bodies, it is essential to know how far away they are; only then can we calculate the sizes and energy outputs of planets, stars, and galaxies, and ultimately determine the scale of the Universe itself. So far as we can tell, mankind has always considered the stars to be further away than the Sun, Moon and planets, but the first measurement of a stellar distance was not until 1837, when Bessel determined the distance to 61 Cygni by TRIGONOMETRICAL PARALLAX.

The method of trigonometrical parallax is similar to the classical techniques used by surveyors, and it is based on the fact that the Earth circles the Sun on an orbit of about 300 million kilometres diameter. As Earth goes on its annual motion (figure 2.2), each of the nearest stars traces out a minute ellipse on the sky with respect to the faraway stars. In the course of a year the apparent position of a nearby star will vary by an angle that we shall here denote as 2p; one half of this angle, p, is termed the PARALLAX of the star, which is normally given in units of seconds of arc (arc sec). In practice the parallax is always less than 1 arc sec. The nearest star, Proxima Centauri, has a parallax of 0.765 arc sec, and 61 Cygni a parallax of 0.293 arc sec; these angles are far less than the smallest discernible with the naked eye. The determination of trigonometrical parallaxes is an exacting and tedious endeavour, requiring many photographs (up to 50) of each star field, spaced over several years. Only a few thousand stars have had their parallaxes measured by trigonometry, and only a few hundred are known correct to 10 per cent.

The method of trigonometrical parallax has given us the impor¬tant unit of distance known as the PARSEC, abbreviated pc. This is the distance at which a star would have a parallax of 1 arc sec, and it is equal to 3.086 X 1013km. In most popular writing the light year is used; there are 3.262 light years in a parsec. Actually there is no star within 1 pc; Proxima Centauri is 1.3pc from us. To turn a parallax quoted in arc seconds into a distance in parsecs it is necessary to divide the quantity 1 by the parallax.

Star distances cannot be found by trigonometrical parallax beyond 300pc as the angle is then immeasurably small, and accurate work is therefore restricted to about 30 pc from the Sun. Consequently if we had only this method at hand the task of surveying our Universe would be impossible. Fortunately we can keep track of distance by other methods.

Another geometrical method exploits the observation that some ot the local star clusters are moving relative to the Sun. All stars in such a cluster (or star family) are travelling on parallel tracks, which, due to perspective effects, appear to converge at one point in the sky. (Likewise the traffic on a straight road appears to converge in the distance.) The distance to a converging cluster is found by comparing its measured angular motion across the sky, in seconds of arc per year, with the true space velocity, in parsecs per year. The angular motion is found by taking’ several photo¬graphs over an interval of years and measuring the systematic changes. To find the true space velocity ^entails measuring the velocity component along the line of sight (radial velocity) and thence across the line of sight, by multiplying the radial velocity by the tangent of the angle between the cluster and the point of convergence. This moving-cluster method has been applied to give accurate results for the Hyades and Ursa Major star clusters, both of which are out of reach of direct surveying.

Apart from these essentially geometrical methods a whole series of distance-measuring techniques in astronomy is based on a simple but elegant principle. We can measure the apparent brightness of a star, and we know that this is determined by three factors: the intrinsic brightness, the distance, and complications such as the presence of dust in space. We will ignore the third while presenting the basic principle. Suppose we measure in magnitudes the apparent brightness m, and can deduce the actual brightness M. Then we can form the quantity (m — M) which, as we shall see in the next section, is linked to distance d (in parsecs) thus:
m — M = 5 Iog10 d — 5
OR
m-M = 51og10(d/10),
where we are taking the actual brightness as the absolute magnitude. In fact there are several classes of stars as well as some galaxies for which we can find the absolute magnitude indirectly and so work out a distance.

The most important of the indirect distance measurements, at least historically, is the method of Cepheid variables. These stars rise and fall in brightness in a regular fashion, and the mean absolute magnitude depends on the time to go through one complete cycle of variability; the longer the period the greater the absolute magnitude. Therefore the distance to a Cepheid variable in a star cluster or another galaxy can be obtained from measurement of the period, which indicates the absolute magnitude, and from measurement of the apparent magnitude.

Another class of variable stars, that of the RR Lyrae variables, also has a link between the period of light variation and absolute magnitude, and so these also are used to map out our Galaxy and its neighbours. Astronomers have also found that for certain exploding stars, namely novae and supernovae, the absolute magnitude at maximum brightness can be deduced from the star’s behaviour. Distances derived from novae and supernovae have an application in the determination of distances beyond the Milky Way.

A further indirect method is that of SPECTROSCOPIC PARALLAX By means of standard techniques in stellar spectroscopy We can discover the physical conditions in a star’s surface and atmosphere Suppose that there exists another star with a similar spectrum and a known distance; we can then argue that both stars are comparable and probably have the same absolute magnitudes. So by relating the two measurable apparent magnitudes for these identical stars and the one known distance we can obtain unknown distance or parallax.

It is important to realise that the measurement of star distances provides the key to the unfolding Universe. The most critical points are concerned with the two geometrical methods, applicable to the nearest stars and moving clusters, and the calibration of the period-luminosity relation for Cepheid variable stars. In the last analysis our ideas of the distances to galaxies and of the size of the Universe itself depend on these particular star-distance methods being right.

We are interested not only in where the stars are now, but also m how they are moving through space. A star’s total space motion can be split into two components: the RADIAL VELOCITY along the line of sight, and the PROPER MOTION at right angles to the line of sight

The radial velocity is found by measuring the Doppler shift of lines in the star’s spectrum. For certain types of star, this wave¬length shift can be found semi-automatically by means of a radial velocity meter; one developed in England at the Cambridge Observatories compares the actual spectrum of a star with that of a similar star at rest and determines the wavelength shift necessary for bringing the two spectra into exact coincidence. From this wavelength shift the velocity is determined by application of the Doppler effect. For most stars it is necessary to measure the observed wavelengths of selected lines on a photographic plate of the spectrum. Radial velocities, which are now known for many thousands of stars, are taken as positive for stars moving away from us, and negative for those traveling towards us.

In principle a single observation of a star discloses the radial velocity, but this is not the case for proper motion. The systematic rate of change of a star’s position against the distant background due to proper motion is usually less than a second of arc per year Consequently it is necessary to take many photographs of the star field, separated by years or decades, in order to get accurate proper motions. It must be recalled that in general only the nearby stars will have measurable proper motions, and these are also the ones that exhibit trigonometrical parallax. A consequence of this is that the derivation of proper motion requires very precise measurement of star positions on photographic plate and computerized analysis of the data.

The star with the largest known proper motion was discovered by the great American observer E E Barnard (1857-1923) Now known as Barnard’s star .It has a proper motion of 10.3 arc sec per year and a parallax of 0.55 arc sec(1.81pc)

By combining information on a star’s distance with its proper motion we obtain the tangential velocity in kilometres per second. Taken in conjunction with the radial velocity, this establishes the true SPACE VELOCITY of the star relative to the Sun. In the solar neighbourhood, most of the space velocities are less than 50km s-1, but these are large enough to cause fundamental changes in the constellation patterns within a few thousand years. Our Sun is also moving along with the restless stars, and its motion through the nearby stars, the direction of the SOLAR APEX, was first worked out by William Herschel in 1783. Our Sun is travelling towards the constellation Hercules at a speed of 20 km s-1.

Comments are closed.